Notes are based on material from the course text, Ref. 1: Introduction to Physical Gas
Dynamics, Vincenti, W. and Kruger, C., Krieger Pub., Copyright 1965. Any figures used from
Ref.1 are so noted and copyrighted by Krieger Publishing Co.

all parts of a system at the same temperature. and composition are macroscopic observable quanti3
.no unbalanced forces. If no body forces. temperature.system has no tendency to undergo a spontaneous
change in chemical composition
If all three of these conditions are satisfied → “Thermodynamic Equilibrium”
Note: pressure.
which is the same temperature as surroundings
Chemical Equilibrium .distribution function: describes probability of an atom or molecule at a
particular point in space having a particular velocity
Boltzmann Equation: PDE which can (in principle) be solved to determine the distribution function
statistical mechanics: describes how energy is partitioned in collections
of atoms and molecules
Macroscopic Description of Gases
Conservation equations for a fluid : these are moments of the Boltzmann
equation
constitutive equation: equation relating stress and strain in the fluid (or
other properties)
equation of state: constitutive equation relating thermodynamic state
functions for a fluid
Equibilibrium of a Thermodynamic System
Mechanical Equilibrium . then
p = const
Thermal Equilibrium .

Local Thermodynamic Equilibrium
(LTE)
.Local Chemical Equilibrium (LCE) implies local chemical composition
corresponds to equilibrium at local temperature and pressure
.implies some subsystem is in local equilibrium internally. though not
necessarily with surroundings. this is the approach taken
in gas dynamics
.e.
These are related to average values of microscopic qantities. we speak of
local equilibrium:
.
Molecular Transport:
mass → diffusion
momentum→ viscosity
energy→ heat conduction
4
.in cases of chemical nonequlibrium.the subsystem can be a fluid element→ often. In a flow. Pressure and Temperature
are related to the distribution of energy at a molecular level. we often still assume local thermal
equilibrium so that temperature can be defined for analysis of reactions
Equilibrium also has implications at a molecular level.ties. i.

C3 and C 2 =
C12 + C22 + C32
. for every
collision deflecting a molecule from its original velocity Ci . there will on average. be
another molecule undergoing a collision resulting in a velocity Ci . C2 . say the wall perpendicular to the
x1 axis:
6
.motion with velocity components C1 . for a small region about a given point.velocity components assumed constant (neglect intermolecular collisions)
The pressure on a wall of this box will be due to force from colliding molecules:
F =
d
(momentum)
dt
We assume the gas molecule collisions are specular
Note: for a gas in equilibrium.
Consider molcules colliding with one of the walls. rotational)
Kinetic Theory: Introduction
Begin with a simple picture to interpret pressure of a gas:
Consider a cubical box of length l on a side:
.molecules in the box are in constant motion
.
⇒ So the gas (in equilibrium) behaves as if the assumption of constant velocity and
specular reflection were true.Internal molecular structure → determines distribution ofenergy in molecule (vibrational.

each with a different mass.two traverses between each collision with wall.and the pressure (force/area) is given by:
mC12 1
mC12
mC12
· 2 = 3 =
p=
l
l
l
V
If there are many molecules..change in momentum per collision:
2m|C1 |
. the pressure on the wall
will be due to the momentum contributions from all:
p=
1 X
mz Cz21
V z
7
. so number of collisions per unit time:
|C1 |
2l
.time to traverse distance l between walls
l
|C1 |
.the total change in momentum per unit time:
2m|C1 | ·
|C1 |
mC12
=
2l
l
.

Comparing the equation of state to the previous expression
for pV
3
Etr = N <T
2
⇒ temperature is a measure of translational kinetic energy
8
.314Jmol−1 K −1 ). ideal gas equation of state can be expressed as
pV = N <T
where N is the number of moles in the system and < is the universal gas constant
(< = 8. These can be combined:
2
pV = Etr
3
Recall that the empirical.There will be pressure exerted on the walls perpendicular to x2 and x3 as well
p=
1 X
1 X
mz (Cz21 + Cz22 + Cz23 ) =
mz Cz2
3V z
3V z
We can also write the translational kinetic energy for the entire system Etr as
Etr =
1X
mz Cz2
2 z
The summation of particle mass times the speed squared is common to both the
pressure and energy.

Note that the Boltzmann constant is related to the universal gas constant and Avogadro’s Number:
k=
<
NA
and Avogadros Number relates the number of molecules
and moles in the system
NA =
Nm
N
so we can write
e˜tr =
Etr
3
= kT
Nm
2
So e˜tr represents the average kinetic energy per molecule.
On a per unit mass basis
etr =
e˜tr
3 kT NA
=
·
mi
2 mi NA
9
.on a per molecule basis
e˜tr =
Etr
Nm
where Nm is the number of molecules in the system.

subscript not shown). Note that M
With these simple results.
ˆ = NA mi .The
internal energy per unit mass can also be written as
etr =
3 <T
3
= RT
ˆ
2M
2
ˆ is the molecular weight of species i and R is
where M
the gas constant for species i on a mass basis (for clarity. we can also find an expression for the specific heats:
Cv =
Cp =
∂e
∂T
∂h
∂T
=
v
p
3
detr
= R
dT
2
5
= R + Cv = R
2
and the ratio of specific heats γ is
γ=
Cp
5
=
Cv
3
This value agrees well with the experimental value for monatomic gases and confirms
validity of simple model which includes only translational kinetic energy.
10
.where mi is the mass of a single molecule of species i.